Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 9.

1. Problem 1.

a) Determine the tangent planes of x^{2} + y^{2} − z^{2} = 1 at the points (x, y, 0) and show that they are all parallel to the z axis.

b)^{∗} Show that the equation of the tangent plane at (x_{o}, y_{o}, z_{o}) of a regular surface given by f(x, y, z) = 0, where 0 is a regular value of f, is

f_{x}(x_{o}, y_{o}, z_{o})(x − x_{o}) + f_{y}(x_{o}, y_{o}, z_{o})(y − y_{o}) + f_{z}(x_{o}, y_{o}, z_{o})(z − z_{o}) = 0.

c) Show that the equation of the tangent plane of a surface which is the graph of a differentiable function z = f(x, y), at the point p_{o} = (x_{o}, y_{o}), is given by

z = f(x_{o}, y_{o}) + f_{x}(x_{o}, y_{o})(x − x_{o}) + f_{y}(x_{o}, y_{o})(y − y_{o}).

Recall the definition of the differential df of a function f : ℝ^{2} → ℝ and show that the tangent plane is the graph of the differential.

2. Problem 2.

If a coordinate neighborhood of a regular surface can be parametrized in the form

r(u, v) = α(u) + β(v),

where α and β are regular parametrized curves, show that the tangent planes along a fixed coordinate curve of this neighborhood are all parallel to a line.

3. Problem 3.

Compute the first fundamental forms of the following parametrized surfaces where they are regular:

a) ellipsoid

r(u, v) = (a sin u cos v, b sin u sin v, c cos u).

b) elliptic paraboloid

r(u, v) = (au cos v, bu sin v, u^{2}).

c) hyperbolic paraboloid

r(u, v) = (au cosh v, bu sinh v, u^{2}).

d) hyperboloid of two sheets

r(u, v) = (a sinh u cos v, b sinh u sin v, c cosh u).

4. Problem 4. a) Given the parametrized surface

r(u, v) = (u cos v, u sin v, log cos v + u), \frac{π}{2} < v < \frac{π}{2} ,

show that the two curves r(u_{1}, v), r(u_{2}, v) determine segments of equal lengths on all curves r(u, const).

b) Show that

r(u, v) = (u sin γ cos v, u sin γ sin v, u cos γ), 0 < u < ∞, 0 < v < 2π, γ = const.,

is a parametrization of the cone with 2γ as the angle of the vertex. In the corresponding coordinate neighborhood, prove that the curve

r(c exp(v sin γ cotan δ), v), c = const, δ = const.,

intersects the generators of the cone (v = const) under the constant angle γ.